In chapter 9 of Genki 1, on page 175, a sample sentence and the translation are given about preserving the tenses when quoting someone.

Here's the starter sentence (what's going to be quoted):
今、日本語を勉強しています。

And here is the quoted sentence with translation.
スーさんは日本語を勉強していると言っていました。
Sue said that she was studying Japanese. (the bolded is what is underlined in the book)

But shouldn't いる be いた if it's talking about what she was doing, not what she is doing right now?
Or, to preserve the tense, shouldn't the translation be is and not was?

This question has been nagging me since this morning when I first noticed it.

My second thought, besides it being a mistake, is that you absolutely can't have a different tense when quoting
someone. But that seems strange though. Let's say Mary called Sue by telephone and Mary said
何をしていますか。and Sue replies 朝ごはんを食べています。. Takeshi asks Mary later in the day what did
Sue say on the telephone, would Mary's reply have to be 朝ごはんを食べていると言っていました。 and not
朝ごはんを食べていたと言っていました。?

The translation in your book uses an indirect quote, which is what's confusing you, I think. In English we have two types of quotes, direct and indirect:

For a direct quote, you can't change the tense:

Sue: I am studying.
You: Sue said "I am studying".

The following would not be acceptable, not matter how much time has passed:
Sue: I am studying.
You: Sue said "I was studying".

For indirect quotes, the tense can match when the action is happening, regardless of when it was said:

10am - Sue: I am studying.
4pm - You: Sue said she was studying.

10am - Sue: I will be studying from 3pm to 7pm.
4pm - You: Sue said she is studying (this evening).

The problem is, grammatically there is not that big of a distinction in Japanese. In writing, if there are quotes (「」) it is a direct quote and you are not supposed to change what the person said. For indirect quotes, you simply remove the 「」, which obviously can't be seen in conversation.

However, the rule is:
The tense of the final verb (言います) dictates when the quote was said, and the tense of the verb before the と (している) dictates when the action happened in relation to the sentence being said.

So, here's your examples with timelines:
スーさんは日本語を勉強していると言っていました。

Far past: Nothing.
Near past: Sue studys and at the same time says 今勉強している。
Now: You say スーさんは日本語を勉強していると言っていました。

So, to answer your first questions, the translation is okay if it is an indirect quote (however, I think changing the "was" to "is" would be fine too!). And for your second question, changing the tense of what was said is not acceptable, because it would mean she was eating before she said the sentence, which is not the case.

Thanks Keatonatron! I think I have a better grasp of the concept now. It's still a little confusing but it's a lot clearer
than it was before.

So the sentence スーさんは日本語を勉強していたと言っていました。could not be feasible if Sue said 今勉強している, but
it would be correct if Sue instead said 勉強していた? And if Sue said 今、朝ごはんを食べている, I would say
スーさんは朝ごはんを食べていると言っていました, correct？

Edit: I think I fully understand it now. When using quotes like this, the verb that has been said by someone,
like 食べている and 勉強している tell the relation between when the speaker said it, and when the speaker
did the action. Like 食べている in 朝ごはんを食べていると言っていました would mean someone was eating breakfast at
the time they said this, while 朝ごはんを食べていたと言っていました would mean someone ate breakfast, but ate
it before they said "I ate breakfast". Is this correct?

TJack wrote:Edit: I think I fully understand it now. When using quotes like this, the verb that has been said by someone,
like 食べている and 勉強している tell the relation between when the speaker said it, and when the speaker
did the action. Like 食べている in 朝ごはんを食べていると言っていました would mean someone was eating breakfast at
the time they said this, while 朝ごはんを食べていたと言っていました would mean someone ate breakfast, but ate
it before they said "I ate breakfast". Is this correct?

tiestotaku wrote:Ok, sorry to jump in- attempting a shot in the dark...

So if the sentence was:

スーさんは彼が勉強していたと言ったと言っていました。

It would mean that Sue said that some guy (or her boyfriend) had said that he was studying, perhaps in the context of you asking about an interruption involving him? ?

Yes. Well, possibly. There are different ways to break it down, depending on the range of the quote:

スーさんは「彼が勉強していたと言った」と言っていました。
Sue was saying "He said he was studying"

or

「スーさんは彼が勉強していたと言った」と言っていました。

Someone was saying "Sue said he he was studying"

If you want to specify that it was Sue who was talking, it's perfectly fine to move the subject (スーさんは) to the end of the sentence (as long as it's before the verb):

彼が勉強していたと言ったと スーさんは言っていました。

Although, in this case I think the は should be changed to が:

彼が勉強していたと言ったと スーさんが言っていました。

The reason for this is that XXXが言いました is the most common form for "s/he said", however when your sentence gets complicated with multiple が's, it's a good idea to change the subject to は to make it more clear (like you did in your sentence). By moving the subject to the end, though, it clears up all confusion over who is doing the final talking, so reverting it to が is preferable.